\begin{abstract}


In  distributed networks, some groups of nodes may have more inter-connections, perhaps due to their larger bandwidth availability or communication requirements. In many scenarios, it may be useful for the nodes to know if they form part of a dense subgraph, e.g., such a dense subgraph could form a high bandwidth backbone for the network.
In this work, we address the problem of self-awareness of nodes in a dynamic network with regards to graph density, i.e., we give distributed algorithms for maintaining dense subgraphs (subgraphs that the member nodes are aware of). The only knowledge that the nodes need is that of the \emph{dynamic diameter} $D$, i.e., the maximum number of rounds it takes for a message to traverse the dynamic network. For our work, we consider a model where the number of nodes are fixed, but a powerful adversary can add or remove a limited number of edges from the network at each time step. The communication is by broadcast only and  follows the CONGEST model in the sense that only messages of $O(\log n)$ size are permitted, where $n$ is the number of nodes in the network.

Our algorithms are continuously executed on the network, and at any time (after some initialization) each node will be aware if it is part (or not) of a particular dense subgraph. We give algorithms that approximate both the \emph{densest subgraph}, i.e., the subgraph of the highest density in the network, and the \emph{at-least-$k$-densest subgraph} (for a given parameter $k$), i.e., the densest subgraph  of size at least $k$. We give a ($2 + \epsilon$)-approximation algorithm for the densest subgraph problem. The at-least-$k$-densest subgraph is known to be NP-hard for the general case in the centralized setting and the best known algorithm gives a 2-approximation. We present an algorithm that maintains a ($3 + \epsilon$)-approximation in our distributed, dynamic setting. Our algorithms run in $O(D\log_{1+\epsilon} n)$ time.

%  where $n$ is the number of nodes in the network.

%Our algorithms rely upon network estimation techniques to get quick and reliable estimation (with suitable probabilistic guarantees) of number of nodes and edges in the network; here, we introduce  a suitable estimation technique, and also note that our algorithms are flexible to be adapted to such models where suitable estimation techniques are available. We then show by clever probabilistic arguments how our algorithms achieve the stated approximation guarantees.


% Maintaining density is a notion that is clearly motivated by the fact that i

%We show that we can maintain an approximated densest subgraph of size at least $k$ in the dynamic networks (edge deletion/addition model).
\end{abstract}
